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How do you prove that every galilean transformation of the space $\mathbb R \times \mathbb R^3$ can be written in a unique way as the composition of a rotation, a translation and uniform motion? Thanks!

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Please explain "uniform motion" and clarify whether you are in 4 dimensions or 3 dimensions or 6 dimensions. – Mark Bennet Aug 21 '12 at 18:07
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@MarkB: Since the question as written refers to galilean transformations of $\Bbb R\times\Bbb R^3$, doesn't it at least imply very strongly that it is considering one time dimension and three spatial? – MJD Aug 21 '12 at 18:15
@MJD - Thanks for clarification ... – Mark Bennet Aug 21 '12 at 18:24
@Mark Bennet: By uniform motion, I mean something of the form g(t,x) = (t,x+v*t), where v is a three-component velocity and x is a vector in R3. – user34801 Aug 21 '12 at 18:37
@user34801 If you don't get a good answer here after a couple of days, you might try asking on physics.stackexchange.com , where people are more likely to have a clue about galilean transformations. – MJD Aug 21 '12 at 18:51
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As mentioned, this problem appears in V.I Arnold's "Mathematical Methods of Classical Mechanics".

A simple proof can be found on page 7 of this nice lecture series on the topic. In essence, you use the affine property of the transformation to show that the time cannot be dilated, the space component must be multiplied by a orthogonal matrix plus a constant vector. The uniqueness follows by considering to different parameterizations and proving equality.

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